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Sharpening the Becker-Stark Inequalities


In this paper, we establish a general refinement of the Becker-Stark inequalities by using the power series expansion of the tangent function via Bernoulli numbers and the property of a function involving Riemann's zeta one.

1. Introduction

Steckin [1] (or see Mitrinovic [2, 3.4.19, page 246]) gives us a result as follows.

Theorem 1.1 (see [1, Lemma ]).

If , then


Later, Becker and Stark [3] (or see Kuang [4, 5.1.102, page 248]) obtain the following two-sided rational approximation for .

Theorem 1.2.

Let , then


Furthermore, and are the best constants in (1.2).

In fact, we can obtain the following further results.

Theorem 1.3.

Let , then


Furthermore, and are the best constants in (1.3).

In this paper, in the form of (1.2) and (1.3) we shall show a general refinement of the Becker-Stark inequalities as follows.

Theorem 1.4.

Let , and let be a natural number. Then


holds, where , and


where are the even-indexed Bernoulli numbers.

Furthermore, and are the best constants in (1.4).

2. Four Lemmas

Lemma 2.1.

The function is decreasing, where is Riemann's zeta function.


is equivalent to the function , which is decreasing.

Lemma 2.2 (see [5, Theorem ]).

Let be Riemann's zeta function and the even-indexed Bernoulli numbers. Then


Lemma 2.3 (see [6, (1.3)]).

Let . Then


Lemma 2.4.

Let and . Then , where



By Lemma 2.3, we have


Since is decreasing by Lemma 2.1, it follows that


From Lemma 2.2, we get


which implies that for .

3. Proofs of Theorems

Proof of Theorem 1.4.





By Lemma 2.4, we have for , and is decreasing on .

At the same time, = by (3.1), and by (3.2), so and are the best constants in (1.4).

Proof of Theorem 1.3.

Let in Theorem 1.4; we obtain that and . Then the proof of Theorem 1.3 is complete.


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Correspondence to Ling Zhu.

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Zhu, L., Hua, J. Sharpening the Becker-Stark Inequalities. J Inequal Appl 2010, 931275 (2010).

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  • Power Series
  • Natural Number
  • Series Expansion
  • Zeta Function
  • Power Series Expansion